The probability that a complete intersection is smooth

Details The probability that a complete intersection is smooth The probability that a complete intersection is smooth

2010-03-26

arxiv.org/abs/1003.5222v2

Mathematics

Number Theory

13 pages; v2: refereed version

Scientific paper

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.

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